The symmetries in the zeros and poles of structured polynomial and rational matrices reflect specific physical properties, as they originate usually from the physical symmetries of the underlying applications [1]}, [2]}, [3]}, [4]}, [5]}. Such special structures occur in numerous applications in engineering, mechanics, control, and linear systems theory. Some of the most common algebraic structures that appear in applications are the (skew-)symmetric (or Hermitian), and the para-(skew-)symmetric (or Hermitian) or alternating structures considered in this work (see [2]}, [4]}, [8]}, [9]} and the references therein). For instance,
symmetric (or Hermitian) matrix polynomials arise in the classical problem of vibration analysis [10]}, [11]}, and alternating matrix polynomials find applications in the study of corner singularities in anisotropic elastic materials [12]} and in the study of gyroscopic systems [3]}. Rational matrices with the structures mentioned above have appeared, for instance, in the continuous-time linear-quadratic optimal control problem and in the spectral factorization problem [1]}, [8]}, [9]}, [17]}.
